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One ofthe most important features of the development of physical
and mathematical sciences in the beginning of the 20th century was
the demolition of prevailing views of the three-dimensional
Euclidean space as the only possible mathematical description of
real physical space. Apriorization of geometrical notions and
identification of physical 3 space with its mathematical modellR
were characteristic for these views. The discovery of non-Euclidean
geometries led mathematicians to the understanding that Euclidean
geometry is nothing more than one of many logically admissible
geometrical systems. Relativity theory amended our understanding of
the problem of space by amalgamating space and time into an
integral four-dimensional manifold. One of the most important
problems, lying at the crossroad of natural sciences and philosophy
is the problem of the structure of the world as a whole. There are
a lot of possibilities for the topology offour dimensional
space-time, and at first sight a lot of possibilities arise in
cosmology. In principle, not only can the global topology of the
universe be complicated, but also smaller scale topological
structures can be very nontrivial. One can imagine two "usual"
spaces connected with a "throat," making the topology of the union
complicated."
This volume provides an introduction to the geometry of manifolds
equipped with additional structures connected with the notion of
symmetry. The content is divided into five chapters. Chapter I
presents the elements of differential geometry which are used in
subsequent chapters. Part of the chapter is devoted to general
topology, part to the theory of smooth manifolds, and the remaining
sections deal with manifolds with additional structures. Chapter II
is devoted to the basic notions of the theory of spaces. One of the
main topics here is the realization of affinely connected symmetric
spaces as totally geodesic submanifolds of Lie groups. In Chapter
IV, the most important classes of vector bundles are constructed.
This is carried out in terms of differential forms. The geometry of
the Euler class is of special interest here. Chapter V presents
some applications of the geometrical concepts discussed. In
particular, an introduction to modern methods of integration of
nonlinear differential equations is given, as well as
considerations involving the theory of hydrodynamic-type Poisson
brackets with connections to interesting algebraic structures. For
mathematicians and mathematical physicists wishing to obtain a good
introduction to the geometry of manifolds. This volume can also be
recommended as a supplementary graduate text.
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